Riemann surfaces have been studied for a long time. Recently, the development of robust conformal mapping algorithms
based on triangulaions has lead to significant interest in discrete versions of smooth theorems and constructions
in the context of Riemann surfaces. In this section I present a selection of images from my research on the topic of
discrete Riemann surfaces.

A Riemann surface can have quite different representations. I consider three kinds of representations in my
work: Embeddings in 3-space, algebraic curves, and quotient spaces of the Euclidean plane, the hyperbolic plane,
and the Riemann sphere. The latter beeing uniformizations of Fuchsian type or schottky type respectively.

Most of the algorithms for the calculation with discrete Riemann surfaces are available online as
part of the VaryLab project.

VaryLab is all about mesh optimization, we say discrete surface optimization. That means
you can modify a given mesh to have minimal energy in a certain sense. The energy in
question is a combination of energies that are defined on the vertex positions of the
input mesh. VaryLab implements various energies for discrete surfaces, e.g., planarity of
faces, equal lengths of edges, curvature of parameter curves and many more.

Discrete Minimal Surfaces
This application calculates discrete minimal surfaces as described in my diploma thesis.
The construction method follows the approach of Bobenko, Hoffmann, and Springborn. It
uses the notion of discrete isothermic surfaces and their Cristoffel transform to define
discrete minimal surfaces. The program is able to create surfaces with planar boundary
curvature lines.

Alexandrov's polyhedron

Alexandrov's Polyhedron
In cooperation with Ivan Izmestiev I implemented an algorithm for constructing
convex polyhedra with a given metric. The java webstart on the left is the main
program which I used for testing and research purposes.

Teamgeist(TM) Polyhedron
The right program is an application of the alexandrov program. It calculates a polyhedron
with predefined combinatorics and symetry of the Teamgeist(TM) soccer ball. I created this
during the world soccer championchips. For a description and a java applet see
A "Teamgeist" Polyhedron

Another polyhedron I created with the help of this tool is the
Reuleaux Triangle Tetrahedron. This is a Tetrahedron with
curved sides which are Reuleaux Triangles. Those triangles get slighly bent to fit together.

The left section is a graph designer. If the graph is
3-connected and enbedded the program calculates the corresponding polyhedron
and displays a normalized representation in the righ section.

This is the first program I created for the geometry group. It contains
numerical algorithms for nonlinear optimization of the convex functional involved.
I use the MTJ
library for linear solving and sparse matrix representation. The program is also part
of my diploma thesis and is being described in detail in Chapter 1.

Private

Piano Playing

Here I publish the piano recordings I'm doing for fun and at home. Feel free to
download and listen. For recording I use a
Seiler 132 Konzert
and the software sample Ivory.